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In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which ...
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
Pages in category "Geometric inequalities" The following 30 pages are in this category, out of 30 total. This list may not reflect recent changes. B.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either or . The figure at the right shows three examples beginning with ...
Erdős–Mordell inequality; Euler's theorem in geometry; Gromov's inequality for complex projective space; Gromov's systolic inequality for essential manifolds; Hadamard's inequality; Hadwiger–Finsler inequality; Hinge theorem; Hitchin–Thorpe inequality; Isoperimetric inequality; Jordan's inequality; Jung's theorem; Loewner's torus ...
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
The proof follows from the arithmetic–geometric mean inequality, , and reciprocal duality (and are also reciprocal dual to each other). The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence ...
In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space. It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log ...